Finite-range lattice DLR existence and closed convex state set

Assume Countable Choice, a finite nonempty alphabet $A$, and a proper consistent finite-range specification $\gamma$ on $X=A^{{\mathbb{Z}}^d}$, with finite range as defined in Lattice Gibbs kernels and DLR states. No translation invariance is required. For every exhaustion ${\mathrm{\Lambda }}_n\uparrow {\mathbb{Z}}^d$ and every sequence of exterior configurations ${\eta }_n$, the measures ${\gamma }_{{\mathrm{\Lambda }}_n}(\cdot \mid {\eta }_n)$ extended with the fixed exterior have a cylinder-convergent subsequence. Every such subsequential limit is a DLR state. Consequently the DLR state set is nonempty, convex, and closed under cylinder convergence. No uniqueness is asserted.

The cited Princeton notes make different broad claims about projective consistency in distinct model sections: the O$(N)$ existence discussion at PDF pp. 16–17 says its finite-volume Gibbs laws do not satisfy the Kolmogorov-extension consistency hypothesis, while the Ising/DLR discussion at pp. 31–32 says the finite-volume family is consistent in that sense. Because those passages concern different model sections, they do not by themselves establish a contradiction for the finite-alphabet specification in this theorem. Neither passage is used here. This proof obtains a DLR state from a cylinder-convergent subsequence, local kernel consistency, and measure extension; it does not treat the finite-volume laws as a projectively consistent family.